Total global neighbourhood domination

Abstract

A subset D of the vertex set of a connected graph G is called a total global neighbourhood dominating set (tgnd-set) of G if and only if D is a total dominating set of G as well as GN, where GN is the neighbourhood graph of G. The total global neighbourhood domination number (tgnd-number) is the minimum cardinality of a total global neighbourhood dominating set of G and is denoted by γtgn(G). In this paper sharp bounds for γtgn are obtained. Exact values of this number for paths and cycles are presented as well. The characterization result for a subset of the vertex set of G to be a total global neighbourhood dominating set for G is given and also characterized the graphs of order n(≥3) having tgnd-numbers 2,n−1,n.